In this talk we deal with the so-called lifting problem on curves. Given an action of
a finite group G over a smooth projective curve in characteristic p, does it always
comes from reduction of an action of G in characteristic zero? It is known that the
answer is yes when (|G|, p) = 1. When wild ramification phenomena appear, the
question becomes much more complex.
In order to study this problem, the notion of Hurwitz tree has been introduced and
successfully exploited in the last ten years. This combinatorial object encodes both
the geometry of fixed points and the ramification theory of the action. We show in
this talk how these Hurwitz trees can be canonically embedded in the Berkovich
unit disk. We will explain how this result sheds new light on the lifting problem and
in which sense these embedded Hurwitz trees "parametrize" certain G-torsors.